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1 Eulerian Polynomials: from Euler s Time to the Present Dominique Foata Dedicated to the memory of Professor Alladi Ramarishnan Abstract. The polynomials commonly called Eulerian today have been introduced by Euler himself in his famous boo Institutiones calculi differentialis cum eius usu in analysi finitorum ac Doctrina serierum (chap. VII, bac in They have been since thoroughly studied, extended, applied. The purpose of the present paper is to go bac to Euler s memoir, find out his motivation and reproduce his derivation, surprisingly partially forgotten. The rebirth of those polynomials in a q-environment is due to Carlitz two centuries after Euler. A brief overview of Carlitz s method is given, as well as a short presentation of combinatorial wors dealing with natural extensions of the classical Eulerian polynomials. 1. Introduction Before Euler s time Jacques Bernoulli had already introduced his famous Bernoulli numbers, denoted by B 2n (n 1 in the sequel. Those numbers can be defined by their generating function as (1.1 u e u 1 = 1 u 2 + n 1 u 2n (2n! ( 1n+1 B 2n, their first values being shown in the table: (1.2 n B 2n 1/6 1/30 1/42 1/30 5/66 691/2730 7/6 Note that besides the first term u/2 there is no term of odd ran in the series expansion (1.1, a property easy to verify. On the other hand, the factor ( 1 n+1 in formula (1.1 and the first values shown in the above table suggest that those numbers are all positive, which is true. Jacques Bernoulli ([Be1713], p had introduced the numbers called after his name to evaluate the sum of the n-th powers of the first m integers. He then proved the following summation formula (1.3 m i n = mn+1 n mn n r n/2 ( n + 1 2r m n 2r+1 ( 1 r+1 B 2r, Invited address at the 10-th Annual Ulam Colloquium, University of Florida, Gainesville, February 18, Key words and phrases. Eulerian polynomials, Bernoulli numbers, Genocchi numbers, tangent numbers, q-eulerian polynomials. Mathematics Subject Classifications. 01A50, 05A15, 05A30, 33B10. 1

2 DOMINIQUE FOATA where n, m 1. Once the first n/2 Bernoulli numbers have been determined (and there are quic ways of getting them, directly derived from (1.1, there are only 2 + n/2 terms to sum on the right-hand side for evaluating m in, whatever the number m. Euler certainly had this summation formula in mind when he looed for an expression for the alternating sum in ( 1 i. Instead of the Bernoulli numbers he introduced another sequence (G 2n (n 1 of integers, later called Genocchi numbers, after the name of Peano s mentor [Ge1852]. They are related to the Bernoulli numbers by the relation (1.4 G 2n := 2(2 2n 1B 2n (n 1, their first values being shown in the next table. n B 2n 1/6 1/30 1/42 1/30 5/66 691/2730 7/6 G 2n Of course, it is not obvious that the numbers G 2n defined by (1.4 are integers and furthermore odd integers. This is a consequence of the little Fermat theorem and the celebrated von Staudt-Clausen theorem (see, for instance, the classical treatise by Nielsen [Ni23] entirely devoted to the studies of Bernoulli numbers and related sequences that asserts that the expression (1.5 ( 1 n B 2n p where the sum is over all prime numbers p such that (p 1 2n, is an integer. From (1.1 and (1.4 we can easily obtain the generating function for the Genocchi numbers in the form: (1.6 2u e u + 1 = u + n 1 1 p, u 2n (2n! ( 1n G 2n. The formula obtained by Euler for the alternating sum m in ( 1 i is quite analogous to Bernoulli s formula (1.3. It suffices to now the first n/2 Genocchi numbers to complete the computation. Euler s formula is the following. Theorem 1.1. Let (G 2n (n 1 be the sequence of numbers defined by relation (1.4 (or by (1.6. If n = 2p 2, then m p ( (1.7 2p ( 1 = ( 1 m m2p 2p 2 + ( 1 m++1 G m2p 2+1, 2

3 EULERIAN POLYNOMIALS while, if n = 2p + 1, the following holds: (1.8 m 2p+1 ( 1 = ( 1 m m2p+1 2 p+1 ( 2p ( 1 m++1 G m2p ( 1 p+1 G 2p+2 4(p + 1. The first values of the numbers G 2n do appear in Euler s memoir. However, he did not bother proving that they were odd integral numbers. The two identities (1.7 and (1.8 have not become classical, in contrast to Bernoulli s formula (1.3, but the effective discovery of the Eulerian polynomials made by Euler for deriving (1.7 and (1.8 has been fundamental in numerous arithmetical and combinatorial studies in modern times. Our purpose in the sequel is to present Euler s discovery by maing a contemporary reading of his calculation. Two centuries after Euler the Eulerian polynomials were given an extension in the algebra of the q-series, thans to Carlitz [Ca54]. Our intention is also to discuss some aspects of that q-extension with a short detour to contemporary wors in Combinatorics. It is a great privilege for me to have met Professor Alladi Ramarishnan, the brilliant Indian physicist and mathematician, who has been influential in so many fields, from Probability to Relativity Theory. He was ind enough to listen to my 2008 University of Florida Ulam Colloquium address and told me of his great admiration for Euler. I am pleased and honored therefore to dedicate the present text to his memory. 2. Euler s definition of the Eulerian polynomials Let (a i (x (i 0 be a sequence of polynomials in the variable x and let t be another variable. For each positive integer m we have the banal identity: (2.1 m 1 i=0 a i (xt i = 1 t m m a i 1 (xt i = a 0 (x + a i (xt i a m (xt m. Now, consider the operator = ( 1 D, where D is the usual! 0 differential operator. Starting with a given polynomial p(x define a i (x := m i p(x (0 i m; m m S(p(x, t := m i p(x t i = a i (x t i. 3

18 DOMINIQUE FOATA For r 1 and each sequence m = (m 1, m 2,..., m r of nonnegative integers let R(m denote the class of all ( m 1 + +m r m 1,...,m r permutations of the multiset 1 m 1 2 m2 r m r. It was already nown and proved by MacMahon [Mac15] that exc and des, on the one hand, maj and inv, on the other hand, were equidistributed on each class R(m, accordingly on each symmetric group S m. However, we had to wait for Riordan [Ri58] for showing that if A n, is defined to be the number of permutations σ from S n having descents (i.e., such that des σ =, then A n, satisfies recurrence (3.5: A n, = ( + 1A, + (n A, 1 (1 n 1; A n,0 = 1 (n 0; A n, = 0 ( n. This result provides the following combinatorial interpretations for the Eulerian polynomials A n (t = σ S n t exc σ = σ S n t des σ, the second equality being due in fact to MacMahon! The latter author [Mac15, p. 97, and p. 186] new how to calculate the generating function for the classes R(m by exc by using his celebrated Master Theorem, but did not mae the connection with the Eulerian polynomials. A thorough combinatorial study of those polynomials was made in the monograph [FS70] in In 1974 Carlitz [Ca74] completes his study of his q-eulerian polynomials by showing that A n (t, q = t des σ q maj σ (n 0. σ S n As inv has the same distribution over S n as maj, it was very tantalizing to mae a full statistical study of the pair (des, inv. Let e q (u := u n /(q; q n be the (first q-exponential. First, a straightforward n 0 calculation leads to the identity 1 + n 1 t A n (t un n! = 1 t 1 t exp((1 tu. In the above fraction mae the substitution exp(u e q (u and express the fraction thereby transformed as a q-series: inv u n 1 t A n (t, q = (q; q n 1 t e q ((1 tu. n 0 The new coefficients inv A n (t, q are to be determined. They were characterized by Stanley [St76] who proved the identity: inv A n (t, q = t t des σ q inv σ (n 1. σ S n 18

19 EULERIAN POLYNOMIALS Now rewrite the exponential generating function for the Eulerian polynomials A n (s (see (3.1 as n 0 A n (s un n! = (1 s exp u exp(su s exp u. In the right-hand side mae the substitutions s sq, exp(u e q (u. Again, express the fraction thereby transformed as a q-series: exc u n A n (s, q = (q; q n n 0 (1 sqe q(u e q (squ sq e q (u. The combinatorial interpretation of the coefficients exc A n (s, q was found by Shareshian and Wachs [SW07] in the form exc A n (s, q = σ S n s exc σ q maj σ (n 0. A further step can be made by calculating the exponential generating function for the polynomials A n (s, t, q := s exc σ t des σ q maj σ (n 0, as was done in [FH08]: σ S n u n A n (s, t, q = (t; q n+1 n 0 r 0 t r (1 sq(usq; q r ((u; q r sq(usq; q r (1 uq r. Identities (4.2 that relate the evaluations of Eulerian polynomials at t = 1 to tangent numbers can also be carried over to a q-environment. This gives rise to a new family of q-analogs of tangent numbers using the combinatorial model of doubloons (see [FH09a], [FH09b]. Following Reiner [Re95a, Re95b] Eulerian polynomials attached to other groups than the symmetric group have been defined and calculated, in particular for Weyl groups. What is needed is the concept of descent, which naturally occurs as soon as the notions of length and positive roots can be introduced. The Eulerian polynomials for Coxeter groups of spherical type have been explicitly calculated by Cohen [Co08], who gave the full answer to a question raised by Hirzebruch [Hi08], who, on the other hand, pointed out the relevance of Euler s memoir [Eul1768] to contemporary Algebraic Geometry. Acnowledgements. The author is most thanful to Hyung Chan Jung (Sogang University, who indly proofread the paper and corrected several typos. 19

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